14 (number)

File:square-pyramidal-14.png.]]

Fourteen is the seventh composite number.

14 is the third distinct semiprime,{{Cite OEIS|sequencenumber=A001358}} being the third of the form $2\; \backslash times\; q$ (where $q$ is a higher prime). More specifically, it is the first member of the second cluster of two discrete semiprimes (14, 15); the next such cluster is (21, 22), members whose sum is the fourteenth prime number, 43.

14 has an aliquot sum of 10, within an aliquot sequence of two composite numbers (14, 10, 8, 7, 1, 0) in the prime 7-aliquot tree.

14 is the third companion Pell number and the fourth Catalan number.{{Cite web|url=https://oeis.org/A002203|title=Sloane's A002203 : Companion Pell numbers|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-06-01}}{{Cite web|url=https://oeis.org/A000108|title=Sloane's A000108 : Catalan numbers|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-06-01}} It is the lowest even $n$ for which the Euler totient $\backslash varphi(x)\; =\; n$ has no solution, making it the first even nontotient.{{Cite web|url=https://oeis.org/A005277|title=Sloane's A005277 : Nontotients|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-06-01}}

According to the Shapiro inequality, 14 is the least number $n$ such that there exist $x\_\{1\}$, $x\_\{2\}$, $x\_\{3\}$, where:{{Cite journal |last=Troesch |first=B. A. |url=https://www.ams.org/journals/mcom/1985-45-171/S0025-5718-1985-0790653-0/S0025-5718-1985-0790653-0.pdf |title=On Shapiro's Cyclic Inequality for N = 13 |journal=Mathematics of Computation |volume=45 |number=171 |date=July 1975 |pages=199 |doi=10.1090/S0025-5718-1985-0790653-0 |doi-access=free |mr=790653 |s2cid=51803624 |zbl=0593.26012 }}

:

with $x\_\{n+1\}\; =\; x\_\{1\}$ and $x\_\{n+2\}\; =\; x\_\{2\}.$

A set of real numbers to which it is applied closure and complement operations in any possible sequence generates 14 distinct sets.{{Cite book |last= Kelley |first= John |authorlink= John L. Kelley |url=https://archive.org/details/generaltopology0000kell |url-access=registration |title= General Topology |publisher= Van Nostrand |location=New York |year= 1955 |page= 57 |isbn= 9780387901251 |oclc=10277303 }} This holds even if the reals are replaced by a more general topological space; see Kuratowski's closure-complement problem.

There are fourteen even numbers that cannot be expressed as the sum of two odd composite numbers:

:$\backslash \{2,\; 4,\; 6,\; 8,\; 10,\; 12,\; 14,\; 16,\; 20,\; 22,\; 26,\; 28,\; 32,\; 38\backslash \}$

where 14 is the seventh such number.{{Cite OEIS |A118081 |Even numbers that can't be represented as the sum of two odd composite numbers. |access-date=2024-08-03 }}

14 is the number of equilateral triangles that are formed by the sides and diagonals of a regular six-sided hexagon.{{Cite OEIS |A238822 |Number of equilateral triangles bounded by the sides and diagonals of a regular 3n-gon. |access-date=2024-05-05 }} In a hexagonal lattice, 14 is also the number of fixed two-dimensional triangular-celled polyiamonds with four cells.{{Cite OEIS |A001420 |Number of fixed 2-dimensional triangular-celled animals with n cells (n-iamonds, polyiamonds) in the 2-dimensional hexagonal lattice. |access-date=2024-05-15 }}

14 is the number of elements in a regular heptagon (where there are seven vertices and edges), and the total number of diagonals between all its vertices.

There are fourteen polygons that can fill a plane-vertex tiling, where five polygons tile the plane uniformly, and nine others only tile the plane alongside irregular polygons.{{Cite journal |first1=Branko |last1=Grünbaum |author-link=Branko Grünbaum |first2=Geoffrey |last2=Shepard |author-link2=G.C. Shephard |title=Tilings by Regular Polygons |date=November 1977 |url=http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |journal=Mathematics Magazine |volume=50 |issue=5 |publisher=Taylor & Francis, Ltd.|page=231 |doi=10.2307/2689529 |jstor=2689529 |s2cid=123776612 |zbl=0385.51006 }}{{Cite web |last=Baez |first=John C. |author-link=John Carlos Baez |url=https://blogs.ams.org/visualinsight/2015/02/01/pentagon-decagon-packing/ |title=Pentagon-Decagon Packing |website=AMS Blogs |publisher=American Mathematical Society |date=February 2015 |access-date=2023-01-18 }}

File:Klein quartic in 14-gon.svg is a regular hyperbolic 14-sided tetradecagon, with an area of $8\backslash pi$.]]

The Klein quartic is a compact Riemann surface of genus 3 that has the largest possible automorphism group order of its kind (of order 168) whose fundamental domain is a regular hyperbolic 14-sided tetradecagon, with an area of $8\backslash pi$ by the Gauss-Bonnet theorem.

Several distinguished polyhedra in three dimensions contain fourteen faces or vertices as facets:

• The cuboctahedron, one of two quasiregular polyhedra, has 14 faces and is the only uniform polyhedron with radial equilateral symmetry.{{Cite book |last=Coxeter |first=H.S.M. |author-link=Harold Scott MacDonald Coxeter |url=https://archive.org/details/regularcomplexpo0000coxe |url-access=registration |title=Regular Polytopes |chapter=Chapter 2: Regular polyhedra |publisher=Dover |edition=3rd |location=New York |year=1973 |pages=18–19 |isbn=0-486-61480-8 |oclc=798003 }}
• The rhombic dodecahedron, dual to the cuboctahedron, contains 14 vertices and is the only Catalan solid that can tessellate space.{{Cite book |last=Williams |first=Robert |chapter-url=https://archive.org/details/geometricalfound0000will/page/168/mode/2up |chapter-url-access=registration |title=The Geometrical Foundation of Natural Structure: A Source Book of Design |chapter=Chapter 5: Polyhedra Packing and Space Filling |publisher=Dover Publications, Inc. |location=New York |year=1979 |pages=168 |isbn=9780486237299 |oclc=5939651 |s2cid=108409770 }}
• The truncated octahedron contains 14 faces, is the permutohedron of order four, and the only Archimedean solid to tessellate space.
• The dodecagonal prism, which is the largest prism that can tessellate space alongside other uniform prisms, has 14 faces.
• The Szilassi polyhedron and its dual, the Császár polyhedron, are the simplest toroidal polyhedra; they have 14 vertices and 14 triangular faces, respectively.{{Cite journal |last= Szilassi |first= Lajos |journal= Structural Topology |pages= 69–80 |title= Regular toroids |url= http://www-iri.upc.es/people/ros/StructuralTopology/ST13/st13-06-a3-ocr.pdf |volume= 13 |year = 1986 |zbl=0605.52002 }}{{Cite journal |last= Császár |first= Ákos |author-link= Ákos Császár |url= http://www.diale.org/pdf/csaszar.pdf |archive-url= https://web.archive.org/web/20170918064243/http://www.diale.org/pdf/csaszar.pdf

| archive-date = 2017-09-18 |title= A polyhedron without diagonals |journal= Acta Scientiarum Mathematicarum (Szeged) |pages= 140–142 |volume= 13 |year= 1949}}

• Steffen's polyhedron, the simplest flexible polyhedron without self-crossings, has 14 triangular faces.{{Cite journal |last1=Lijingjiao |first1=Iila |last2=Tachi |first2=Tomohiro |last3=Guest |first3=Simon D. |display-authors=1 |url=https://www.repository.cam.ac.uk/bitstream/handle/1810/279138/IASS2015_paper%20Author%20Accepted%20Version.pdf?sequence=1&isAllowed=y |title=Optimizing the Steffen flexible polyhedron |journal=Proceedings of the International Association for Shell and Spatial Structures (Future Visions Symposium) |location=Amsterdam |publisher=IASS |date=2015 |doi=10.17863/CAM.26518 |s2cid=125747070 }}

A regular tetrahedron cell, the simplest uniform polyhedron and Platonic solid, is made up of a total of 14 elements: 4 edges, 6 vertices, and 4 faces.

• Szilassi's polyhedron and the tetrahedron are the only two known polyhedra where each face shares an edge with each other face, while Császár's polyhedron and the tetrahedron are the only two known polyhedra with a continuous manifold boundary that do not contain any diagonals.
• Two tetrahedra that are joined by a common edge whose four adjacent and opposite faces are replaced with two specific seven-faced crinkles will create a new flexible polyhedron, with a total of 14 possible clashes where faces can meet.{{cite thesis |last=Li |first=Jingjiao |year=2018 |title=Flexible Polyhedra: Exploring finite mechanisms of triangulated polyhedra |url=https://core.ac.uk/download/pdf/151178534.pdf |type=Ph.D. Thesis |publisher=University of Cambridge, Department of Engineering |pages= xvii, 1-156 |doi=10.17863/CAM.18803 |doi-access=free |s2cid=204175310 }}^pp.10-11,14 This is the second simplest known triangular flexible polyhedron, after Steffen's polyhedron.^p.16 If three tetrahedra are joined at two separate opposing edges and made into a single flexible polyhedron, called a 2-dof flexible polyhedron, each hinge will only have a total range of motion of 14 degrees.^p.139

14 is also the root (non-unitary) trivial stella octangula number, where two self-dual tetrahedra are represented through figurate numbers, while also being the first non-trivial square pyramidal number (after 5);{{Cite OEIS |A007588 |Stella octangula numbers |access-date=2023-01-18 }}{{Cite OEIS |A000330 |Square pyramidal numbers |access-date=2023-01-18 }} the simplest of the ninety-two Johnson solids is the square pyramid $J\_\{1\}.${{efn|1=Furthermore, the square pyramid can be attached to uniform and non-uniform polyhedra (such as other Johnson solids) to generate fourteen other Johnson solids: J₈, J₁₀, J₁₅, J₁₇, J₄₉, J₅₀, J₅₁, J₅₂, J₅₃, J₅₄, J₅₅, J₅₆, J₅₇, and J₈₇. }} There are a total of fourteen semi-regular polyhedra, when the pseudorhombicuboctahedron is included as a non-vertex transitive Archimedean solid (a lower class of polyhedra that follow the five Platonic solids).{{Cite journal |last=Grünbaum |first=Branko |author-link=Branko Grünbaum |title=An enduring error |url=https://ems.press/content/serial-article-files/2342 |journal=Elemente der Mathematik |volume=64 |issue=3 |pages=89–101 |publisher=European Mathematical Society |location=Helsinki |year=2009 |doi=10.4171/EM/120 |doi-access=free |mr=2520469 |zbl=1176.52002 |s2cid=119739774 }}{{Cite journal |last1=Hartley |first1=Michael I. |last2=Williams |first2= Gordon I. |title=Representing the sporadic Archimedean polyhedra as abstract polytopes |journal=Discrete Mathematics |volume=310 |issue=12 |publisher=Elsevier |location=Amsterdam |pages=1835–1844 |year=2010 |doi=10.1016/j.disc.2010.01.012 |doi-access=free |arxiv=0910.2445 |bibcode=2009arXiv0910.2445H |mr=2610288 |zbl=1192.52018 |s2cid=14912118 }}{{efn|1=Where the tetrahedron — which is self-dual, inscribable inside all other Platonic solids, and vice versa — contains fourteen elements, there exist thirteen uniform polyhedra that contain fourteen faces (U₀₉, U_76i, U₀₈, U_77c, U₀₇), vertices (U_76d, U_77d, U_78b, U_78c, U_79b, U_79c, U_80b) or edges (U₁₉). }}

Fourteen possible Bravais lattices exist that fill three-dimensional space.{{Cite OEIS |A256413 |Number of n-dimensional Bravais lattices. |access-date=2023-01-18 }}

The exceptional Lie algebra G₂ is the simplest of five such algebras, with a minimal faithful representation in fourteen dimensions. It is the automorphism group of the octonions $\backslash mathbb\; \{O\}$, and holds a compact form homeomorphic to the zero divisors with entries of unit norm in the sedenions, $\backslash mathbb\; \{S\}$.{{Cite journal | last1 = Baez | first1 = John C. | author-link = John Baez | title = The Octonions | journal = Bulletin of the American Mathematical Society |series = New Series | volume = 39 | issue = 2 | page = 186 | year = 2002 | url = https://math.ucr.edu/home/baez/octonions/node14.html | doi = 10.1090/S0273-0979-01-00934-X | arxiv = math/0105155| mr = 1886087 | s2cid = 586512 | zbl = 1026.17001 }}{{Citation | last1=Moreno | first1=Guillermo | title=The zero divisors of the Cayley–Dickson algebras over the real numbers | year=1998 | journal= Bol. Soc. Mat. Mexicana |series=Series 3| volume=4 | issue=1 | pages=13–28 | arxiv=q-alg/9710013 | bibcode=1997q.alg....10013G | mr=1625585 | s2cid = 20191410 | zbl = 1006.17005 }}

The floor of the imaginary part of the first non-trivial zero in the Riemann zeta function is $14$,{{Cite OEIS |A013629 |Floor of imaginary parts of nontrivial zeros of Riemann zeta function. |access-date=2024-01-16 }} in equivalence with its nearest integer value,{{Cite OEIS |A002410 |Nearest integer to imaginary part of n-th zero of Riemann zeta function. |access-date=2024-01-16 }} from an approximation of $14.1347251417\backslash ldots${{Cite OEIS |A058303 |Decimal expansion of the imaginary part of the first nontrivial zero of the Riemann zeta function. |access-date=2024-01-16 }}{{Cite web |last=Odlyzko |first=Andrew |author-link=Andrew Odlyzko |title=The first 100 (non trivial) zeros of the Riemann Zeta function [AT&T Labs]|url=https://www-users.cse.umn.edu/~odlyzko/zeta_tables/zeros2 |website=Andrew Odlyzko: Home Page |publisher=UMN CSE |access-date=2024-01-16 }}

There is a fourteen-point silver star marking the traditional spot of Jesus’ birth in the Basilica of the Nativity in Bethlehem. According to the genealogy of Jesus in the Gospel of Matthew, “…there were fourteen generations in all from Abraham to David, fourteen generations from David to the exile to Babylon, and fourteen from the exile to the Messiah” (Matthew 1:17).

Islam

In Islam, 14 has a special significance because of the Fourteen Infallibles who are especially revered and important in Twelver Shi'ism. They are all considered to be infallible by Twelvers alongside the Prophets of Islam, however these fourteen are said to have a greater significance and closeness to God.

These fourteen include:

1. Prophet Muhammad (SAWA)
2. His daughter, Lady Fatima (SA)
3. Her husband, Imam Ali (AS)
4. His son, Imam Hasan (AS)
5. His brother, Imam Husayn (AS)
6. His son, Imam Ali al-Sajjad (AS)
7. His son, Imam Muhammad al-Baqir (AS)
8. His son, Imam Ja'far al-Sadiq (AS)
9. His son, Imam Musa al-Kazim (AS)
10. His son, Imam Ali al-Rida (AS)
11. His son, Imam Muhammad al-Jawad (AS)
12. His son, Imam Ali al-Hadi (AS)
13. His son, Imam Hasan al-Askari (AS)
14. His son, Imam Muhammad al-Mahdi (AJTFS)

The number 14 was linked to Šumugan and Nergal.{{sfn|Wiggermann|1998|p=222}}

Fourteen is:

• The number of days in a fortnight.

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• {{citation|first=Frans A. M.|last=Wiggermann|entry=Nergal A. Philological|encyclopedia=Reallexikon der Assyriologie|entry-url=http://publikationen.badw.de/en/rla/index#8358|year=1998|access-date=2022-03-06}}

{{Integers|zero}}

Category:Integers